Journal Article

Hopf bifurcation in epidemic models with a time delay in vaccination

Q. J. A. KHAN and DAVID GREENHALGH

in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 16, issue 2, pages 113-142
Published in print June 1999 | ISSN: 1477-8599
Published online June 1999 | e-ISSN: 1477-8602 | DOI: http://dx.doi.org/10.1093/imammb/16.2.113
Hopf bifurcation in epidemic models with a time delay in vaccination

More Like This

Show all results sharing these subjects:

  • Applied Mathematics
  • Biomathematics and Statistics

GO

Show Summary Details

Preview

Two SIR models for the spread of infectious diseases which were originally suggested by Greenhalgh & Das (1995, Theor Popul. Biol. 47, 129–179; 1995, Mathematical Population Dynamics: Analysis of Hetrerogeneity, pp. 79–101, Winnipeg: Wuerz Publishing) are considered but with a time delay in the vaccination term. This reflects the fact that real vaccines do not immediately confer permanent immunity. The population is divided into susceptible, infectious, and immune classes. The contact rate is constant in model I but it depends on the population size in model II. The death rate depends on the population size in both models. There is an additional mortality due to the disease, and susceptibles are vaccinated and may become permanently immune after a lapse of some time. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that that for diseases in human populations Hopf bifurcation is unlikely to occur at realistic parameter values if the death rate is a concave function of the population size.

Keywords: epidemic model; immunization; time delay; density dependence; Hopf bifurcation; differential equations

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

Full text: subscription required

How to subscribe Recommend to my Librarian

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.